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February  2016, 36(2): 971-980. doi: 10.3934/dcds.2016.36.971

## Topological degree method for the rotationally symmetric $L_p$-Minkowski problem

 1 Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China 2 Department of Mathematical Sciences, Tsinghua University, Beijing 100084

Received  October 2014 Revised  February 2015 Published  August 2015

Consider the existence of rotationally symmetric solutions to the $L_p$-Minkowski problem for $p=-n-1$. Recently a sufficient condition was obtained for the existence via the variational method and a blow-up analysis in [16]. In this paper we use a topological degree method to prove the same existence and show the result holds under a similar complementary sufficient condition. Moreover, by this degree method, we obtain the existence result in a perturbation case.
Citation: Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971
##### References:
 [1] J. Ai, K.-S. Chou and J.-C. Wei, Self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var. Partial Differential Equations, 13 (2001), 311.  doi: 10.1007/s005260000075.  Google Scholar [2] L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing,, Arch. Rational Mech. Anal., 123 (1993), 199.  doi: 10.1007/BF00375127.  Google Scholar [3] B. Andrews, Evolving convex curves,, Calc. Var. Partial Differential Equations, 7 (1998), 315.  doi: 10.1007/s005260050111.  Google Scholar [4] J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem,, J. Amer. Math. Soc., 26 (2013), 831.  doi: 10.1090/S0894-0347-2012-00741-3.  Google Scholar [5] E. Calabi, Complete affine hyperspheres. I,, in Symposia Mathematica, (1972), 19.   Google Scholar [6] S.-Y. A. Chang, M. J. Gursky and P. C. Yang, The scalar curvature equation on $2$- and $3$-spheres,, Calc. Var. Partial Differential Equations, 1 (1993), 205.  doi: 10.1007/BF01191617.  Google Scholar [7] W.-X. Chen, $L_p$ Minkowski problem with not necessarily positive data,, Adv. Math., 201 (2006), 77.  doi: 10.1016/j.aim.2004.11.007.  Google Scholar [8] W.-X. Chen and C.-M. Li, A necessary and sufficient condition for the Nirenberg problem,, Comm. Pure Appl. Math., 48 (1995), 657.  doi: 10.1002/cpa.3160480606.  Google Scholar [9] K.-S. Chou and X.-J. Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry,, Adv. Math., 205 (2006), 33.  doi: 10.1016/j.aim.2005.07.004.  Google Scholar [10] K.-S. Chou and X.-P. Zhu, The Curve Shortening Problem,, Chapman & Hall/CRC, (2001).  doi: 10.1201/9781420035704.  Google Scholar [11] J.-B. Dou and M.-J. Zhu, The two dimensional $L_p$ Minkowski problem and nonlinear equations with negative exponents,, Adv. Math., 230 (2012), 1209.  doi: 10.1016/j.aim.2012.02.027.  Google Scholar [12] M. Ji, On positive scalar curvature on $S^2$,, Calc. Var. Partial Differential Equations, 19 (2004), 165.  doi: 10.1007/s00526-003-0214-0.  Google Scholar [13] H.-Y. Jian and X.-J. Wang, Bernsterin theorem and regularity for a class of Monge Ampère equations,, J. Diff. Geom., 93 (2013), 431.   Google Scholar [14] M.-Y. Jiang, L.-P. Wang and J.-C. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var. Partial Differential Equations, 41 (2011), 535.  doi: 10.1007/s00526-010-0375-6.  Google Scholar [15] Y.-Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I,, J. Differential Equations, 120 (1995), 319.  doi: 10.1006/jdeq.1995.1115.  Google Scholar [16] J. Lu and X.-J. Wang, Rotationally symmetric solutions to the $L_p$-Minkowski problem,, J. Differential Equations, 254 (2013), 983.  doi: 10.1016/j.jde.2012.10.008.  Google Scholar [17] E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem,, J. Differential Geom., 38 (1993), 131.   Google Scholar [18] E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem,, Trans. Amer. Math. Soc., 356 (2004), 4359.  doi: 10.1090/S0002-9947-03-03403-2.  Google Scholar [19] E. Lutwak and G. Zhang, Blaschke-Santaló inequalities,, J. Diff. Geom., 47 (1997), 1.   Google Scholar [20] R. Schoen and D. Zhang, Prescribed scalar curvature on the $n$-sphere,, Calc. Var. Partial Differential Equations, 4 (1996), 1.  doi: 10.1007/BF01322307.  Google Scholar [21] G. Szego, Orthogonal Polynomials,, American Mathematical Society, (1975).   Google Scholar [22] V. Umanskiy, On solvability of two-dimensional $L_p$-Minkowski problem,, Adv. Math., 180 (2003), 176.  doi: 10.1016/S0001-8708(02)00101-9.  Google Scholar [23] G. Zhu, The logarithmic Minkowski problem for polytopes,, Adv. Math., 262 (2014), 909.  doi: 10.1016/j.aim.2014.06.004.  Google Scholar [24] G. Zhu, The centro-affine Minkowski problem for polytopes,, J. Differential Geom., 101 (2015), 159.   Google Scholar

show all references

##### References:
 [1] J. Ai, K.-S. Chou and J.-C. Wei, Self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var. Partial Differential Equations, 13 (2001), 311.  doi: 10.1007/s005260000075.  Google Scholar [2] L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing,, Arch. Rational Mech. Anal., 123 (1993), 199.  doi: 10.1007/BF00375127.  Google Scholar [3] B. Andrews, Evolving convex curves,, Calc. Var. Partial Differential Equations, 7 (1998), 315.  doi: 10.1007/s005260050111.  Google Scholar [4] J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The logarithmic Minkowski problem,, J. Amer. Math. Soc., 26 (2013), 831.  doi: 10.1090/S0894-0347-2012-00741-3.  Google Scholar [5] E. Calabi, Complete affine hyperspheres. I,, in Symposia Mathematica, (1972), 19.   Google Scholar [6] S.-Y. A. Chang, M. J. Gursky and P. C. Yang, The scalar curvature equation on $2$- and $3$-spheres,, Calc. Var. Partial Differential Equations, 1 (1993), 205.  doi: 10.1007/BF01191617.  Google Scholar [7] W.-X. Chen, $L_p$ Minkowski problem with not necessarily positive data,, Adv. Math., 201 (2006), 77.  doi: 10.1016/j.aim.2004.11.007.  Google Scholar [8] W.-X. Chen and C.-M. Li, A necessary and sufficient condition for the Nirenberg problem,, Comm. Pure Appl. Math., 48 (1995), 657.  doi: 10.1002/cpa.3160480606.  Google Scholar [9] K.-S. Chou and X.-J. Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry,, Adv. Math., 205 (2006), 33.  doi: 10.1016/j.aim.2005.07.004.  Google Scholar [10] K.-S. Chou and X.-P. Zhu, The Curve Shortening Problem,, Chapman & Hall/CRC, (2001).  doi: 10.1201/9781420035704.  Google Scholar [11] J.-B. Dou and M.-J. Zhu, The two dimensional $L_p$ Minkowski problem and nonlinear equations with negative exponents,, Adv. Math., 230 (2012), 1209.  doi: 10.1016/j.aim.2012.02.027.  Google Scholar [12] M. Ji, On positive scalar curvature on $S^2$,, Calc. Var. Partial Differential Equations, 19 (2004), 165.  doi: 10.1007/s00526-003-0214-0.  Google Scholar [13] H.-Y. Jian and X.-J. Wang, Bernsterin theorem and regularity for a class of Monge Ampère equations,, J. Diff. Geom., 93 (2013), 431.   Google Scholar [14] M.-Y. Jiang, L.-P. Wang and J.-C. Wei, $2\pi$-periodic self-similar solutions for the anisotropic affine curve shortening problem,, Calc. Var. Partial Differential Equations, 41 (2011), 535.  doi: 10.1007/s00526-010-0375-6.  Google Scholar [15] Y.-Y. Li, Prescribing scalar curvature on $S^n$ and related problems. I,, J. Differential Equations, 120 (1995), 319.  doi: 10.1006/jdeq.1995.1115.  Google Scholar [16] J. Lu and X.-J. Wang, Rotationally symmetric solutions to the $L_p$-Minkowski problem,, J. Differential Equations, 254 (2013), 983.  doi: 10.1016/j.jde.2012.10.008.  Google Scholar [17] E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem,, J. Differential Geom., 38 (1993), 131.   Google Scholar [18] E. Lutwak, D. Yang and G. Zhang, On the $L_p$-Minkowski problem,, Trans. Amer. Math. Soc., 356 (2004), 4359.  doi: 10.1090/S0002-9947-03-03403-2.  Google Scholar [19] E. Lutwak and G. Zhang, Blaschke-Santaló inequalities,, J. Diff. Geom., 47 (1997), 1.   Google Scholar [20] R. Schoen and D. Zhang, Prescribed scalar curvature on the $n$-sphere,, Calc. Var. Partial Differential Equations, 4 (1996), 1.  doi: 10.1007/BF01322307.  Google Scholar [21] G. Szego, Orthogonal Polynomials,, American Mathematical Society, (1975).   Google Scholar [22] V. Umanskiy, On solvability of two-dimensional $L_p$-Minkowski problem,, Adv. Math., 180 (2003), 176.  doi: 10.1016/S0001-8708(02)00101-9.  Google Scholar [23] G. Zhu, The logarithmic Minkowski problem for polytopes,, Adv. Math., 262 (2014), 909.  doi: 10.1016/j.aim.2014.06.004.  Google Scholar [24] G. Zhu, The centro-affine Minkowski problem for polytopes,, J. Differential Geom., 101 (2015), 159.   Google Scholar
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